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BIOINFORMATICS
Vol.28 ISMB 2012,pages i283–i291
doi:10.1093/bioinformatics/bts225
Efﬁcient algorithms for the reconciliation problem with gene
duplication,horizontal transfer and loss
Mukul S.Bansal
1,∗
,Eric J.Alm
2
and Manolis Kellis
1,3,∗
1
Computer Science and Artiﬁcial Intelligence Laboratory,
2
Department of Biological Engineering,Massachusetts
Institute of Technology,Cambridge,MA 02139,USA and
3
Broad Institute of MIT and Harvard,Cambridge,MA
02142,USA
ABSTRACT
Motivation:Gene family evolution is driven by evolutionary events
such as speciation,gene duplication,horizontal gene transfer and
gene loss,and inferring these events in the evolutionary history
of a given gene family is a fundamental problem in comparative
and evolutionary genomics with numerous important applications.
Solving this problem requires the use of a reconciliation framework,
where the input consists of a gene family phylogeny and the
corresponding species phylogeny,and the goal is to reconcile the
two by postulating speciation,gene duplication,horizontal gene
transfer and gene loss events.This reconciliation problemis referred
to as duplicationtransferloss (DTL) reconciliation and has been
extensively studied in the literature.Yet,even the fastest existing
algorithms for DTL reconciliation are too slow for reconciling large
gene families and for use in more sophisticated applications such as
gene tree or species tree reconstruction.
Results:We present two new algorithms for the DTL reconciliation
problem that are dramatically faster than existing algorithms,both
asymptotically and in practice.We also extend the standard DTL
reconciliation model by considering distancedependent transfer
costs,which allow for more accurate reconciliation and give
an efﬁcient algorithm for DTL reconciliation under this extended
model.We implemented our new algorithms and demonstrated
up to 100000fold speedup over existing methods,using both
simulated and biological datasets.This dramatic improvement
makes it possible to use DTL reconciliation for performing rigorous
evolutionary analyses of large gene families and enables its use in
advanced reconciliationbased gene and species tree reconstruction
methods.
Availability:Our programs can be freely downloaded from
http://compbio.mit.edu/rangerdtl/.
Contact:mukul@csail.mit.edu;manoli@mit.edu
Supplementary information:Supplementary data are available at
Bioinformatics online.
1 INTRODUCTION
Gene families evolve through complex evolutionary processes such
as speciation,gene duplication,horizontal gene transfer and gene
loss.Accurate inference of these events is crucial not only to
understanding gene and genome evolution but also for reliably
inferring orthologs,paralogs,and xenologs (Koonin,2005;Mi et al.,
2010;Sennblad and Lagergren,2009;Storm and Sonnhammer,
2002;van der Heijden et al.,2007;Vilella et al.,2009;Wapinski
∗
To whom correspondence should be addressed.
et al.,2007);reconstructing ancestral gene content and dating
gene birth (Chen et al.,2000;David and Alm,2011;Ma et al.,
2008);accurate gene tree reconstruction (Rasmussen and Kellis,
2011;Vilella et al.,2009);and for whole genome species tree
reconstruction (Bansal et al.,2007;Burleigh et al.,2011).Indeed,
the problemof inferring gene family evolution has been extensively
studied.In the typical formulation of this problem,the goal
is to reconcile an input gene tree (gene family phylogeny) to
the corresponding rooted species tree by postulating speciation,
duplication,transfer and loss events.Much of the previous work
in gene tree–species tree reconciliation has focused on either
duplication loss (DL) (Bonizzoni et al.,2005;Chauve et al.,2008;
Durand et al.,2006;Eulenstein and Vingron,1998;Goodman et al.,
1979;Górecki and Tiuryn,2006;Mirkin et al.,1995;Page,1994) or
transfer loss (TL) (Boc et al.,2010;Hallett and Lagergren,2001;Hill
et al.,2010;Huelsenbeck et al.,2000;Jin et al.,2009;Nakhleh et al.,
2004,2005;Ronquist,1995),but not on duplication,transfer and
loss together.However,duplication and transfer events frequently
occur together,particularly in prokaryotes,and the analysis of such
families requires reconciliation methods that can simultaneously
consider duplication,transfer and loss events.This problem of
gene tree–species tree reconciliation by duplication,transfer and
loss simultaneously is referred to as the duplication TL (DTL)
reconciliation problem.
Previous work.The DTL reconciliation problem has a long history
and is well studied in the literature.This is partly due to its close
association with the host–parasite cophylogeny problem(Charleston
and Perkins,2006) which seeks to understand the evolution of
parasites (analogous to genes) within hosts (analogous to species).
Almost all known formulations of the DTL reconciliation problem
are based on a parsimony framework (Charleston,1998;Conow
et al.,2010;David andAlm,2011;Doyon et al.,2010;Gorbunov and
Liubetskii,2009;LibeskindHadas and Charleston,2009;Merkle
and Middendorf,2005;Merkle et al.,2010;Ovadia et al.,2011;
Ronquist,2003;Toﬁgh et al.,2011) (but see also Toﬁgh (2009)
for an example of a probabilistic formulation,as well as Csürös
and Miklós (2006) for a probabilistic framework based on gene
content).Under this framework,each duplication,transfer and loss
event is assigned a cost and the goal is to ﬁnd a reconciliation that
has the lowest total reconciliation cost.Optimal DTLreconciliations
can sometimes violate temporal constraints;that is,the transfers
are such that they induce contradictory constraints on the dates for
the internal nodes of the species tree.We refer to such paradoxical
reconciliations as timeinconsistent (after Doyon et al.,2010).In
general,it is desirable to consider only those DTL reconciliations
that are timeconsistent (i.e.paradoxfree).Henceforth,we refer
to the problem of speciﬁcally computing optimal timeconsistent
© The Author(s) 2012.Published by Oxford University Press.
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M.S.Bansal et al.
DTLreconciliations as the tcDTL Reconciliation problem.Although
the DTL reconciliation problem can be solved in polynomial time,
solving the tcDTLreconciliation problemis NPhard (Ovadia et al.,
2011;Toﬁgh et al.,2011).If divergence time information is available
for the nodes of the species tree (or if there is a known relative
temporal ordering for each pair of internal nodes),then any proposed
DTL reconciliation must also respect the temporal constraints
imposed by the available timing information,i.e.,transfers must
be restricted to occur only between coexisting species.When
such divergence timing information is available,even the tcDTL
reconciliation problem becomes polynomially solvable (Libeskind
Hadas and Charleston,2009).(Note,however,that timeconsistency
cannot be guaranteed just by ensuring that transfers occur between
coexisting species).In general,the input species tree can be undated,
partially dated,or fully dated depending on whether the divergence
timing information associated with the nodes of the species tree is
absent,partial,or complete,respectively.Thus,in practice,when
the species tree is undated or partially dated,one solves the DTL
reconciliation problem,and if the species tree is fully dated,one
can solve either the DTL reconciliation or the tcDTL reconciliation
problem.
Let mdenote the number of leaves in the input gene tree and n the
number of leaves in the species tree.Both the DTL reconciliation
problem and the tcDTL reconciliation problem,along with some
of their variants,have been extensively studied in the literature
(Charleston,1998;Conowet al.,2010;David andAlm,2011;Doyon
et al.,2010;Gorbunov and Liubetskii,2009;LibeskindHadas and
Charleston,2009;Merkle and Middendorf,2005;Merkle et al.,
2010;Ronquist,2003;Toﬁgh,2009;Toﬁgh et al.,2011).The most
recent algorithmic work on these problems includes Doyon et al.
(2010);Toﬁgh (2009);Toﬁgh et al.(2011);and David and Alm,
2011.The paper by Toﬁgh et al.(2011) studies a restricted version of
the reconciliation model that ignores losses (equivalent to assigning
a cost of zero for loss events under the DTLreconciliation problem)
and shows that,under this restricted model,the DTL reconciliation
problem on undated trees can be solved in O(mn
2
) time.They also
gave a ﬁxed parameter tractable algorithmfor enumerating all most
parsimonious reconciliations.The time complexity of the O(mn
2
)
time algorithm was further improved to O(mn) in Toﬁgh (2009)
(under the same restricted reconciliation model).However,with the
increasing availability of wholegenome datasets,such a restriction
on the reconciliation model can be problematic as losses are a rich
source of information that can be critical for accurate reconciliation.
Indeed,losses play a fundamental role in the ability to distinguish
between duplications and transfers as well as in mapping the nodes
of the gene tree into the nodes of the species tree,and thus should
be explicitly considered during reconciliation.The paper by Doyon
et al.(2010) showed that,for fully dated species trees,the tcDTL
reconciliation problem could be solved in O(mn
2
) time.Recently,
an O(mn
2
)time algorithm for the tcDTL reconciliation problem on
fully dated trees has also been independently developed for Version 2
of the program Jane (Conow et al.,2010).Finally,the recent paper
by David and Alm (2011) gave an O(mn
2
)time algorithm for the
DTL reconciliation problem on undated trees.
In summary,in spite of tremendous methodological and
algorithmic advances,even the fastest existing algorithms for DTL
reconciliation (David and Alm,2011;Merkle et al.,2010) as well
as for tcDTLreconciliation on fully dated trees (Doyon et al.,2010)
still have a time complexity of (mn
2
).This makes themtoo slowto
reconcile trees with more than a few hundred taxa,and completely
unsuitable for all but the smallest trees when used in sophisticated
applications such as reconciliationbased gene tree or species tree
reconstruction that require the reconciliation of a multitude of trees
while searching through tree space (Bansal et al.,2007;Burleigh
et al.,2011;Rasmussen and Kellis,2011;Vilella et al.,2009).
Our contributions.Recall that the DTLreconciliation problem,even
on fully dated species trees,does not guarantee that the optimal
reconciliation is timeconsistent,whereas the tcDTL reconciliation
problem does.However,the tcDTL reconciliation problem suffers
from two major drawbacks that limit its applicability in practice.
First,the tcDTL reconciliation problem can only be solved
efﬁciently when the species tree is fully dated.This limits its
application to only those species tree that contain a relatively small
number of taxa (say <100).This is because,it can be extremely
difﬁcult to accurately date large species trees (Rutschmann,2006)
and the accuracy of tcDTLreconciliation relies implicitly on having
a correctly dated species tree.Second,the time complexity of the
fastest known algorithm for the tcDTL reconciliation problem is
O(mn
2
),which makes it too slow to be used with large datasets
(as we also demonstrate later).This also makes it too slow for
reconciliationbased gene tree reconstruction of even relatively
small gene trees,as it involves repeatedly reconciling a multitude
of candidate gene trees against the species tree.Furthermore,the
tcDTL reconciliation problem cannot be used for reconciliation
based wholegenome species tree construction (also called gene tree
parsimony),as the topology of the species tree is repeatedly modiﬁed
and so at each step,the species tree is undated.
Thus,in this work,we focus on the DTL reconciliation problem.
In particular,we improve upon the current state of the art for the
DTL reconciliation problem in the following ways:
(1) We provide an O(mn)time algorithm for the DTL
reconciliation problem on undated species trees.This
improves on the fastest known algorithm for this problem
by a factor of n.The DTL reconciliation problem on undated
trees is the most common version of the DTL reconciliation
problem and arises whenever the species tree cannot be
accurately dated,as is usually the case with large gene
families,and in applications such as reconciliationbased
species tree reconstruction.
(2) For the DTL reconciliation problem on fully dated species
trees,we provide an O(mn logn)time algorithm,which
improves on the fastest known algorithm for this problem
by a factor of n/logn.Even though the fully dated version
of DTL reconciliation does not guarantee timeconsistency,
as we show later using thorough experimental studies,
optimal DTL reconciliations closely approximate optimal
tcDTL reconciliations.This algorithm is thus meant as a
faster alternative to the O(mn
2
)time algorithm for tcDTL
reconciliation.
(3) We give a simple O(mn
2
)time algorithm for DTL
reconciliation that can handle distancedependent transfer
costs and can work with undated,partially dated,or fully
dated species trees.This is a factor of n faster than the fastest
known algorithm that can handle distancedependent transfer
costs (Conow et al.,2010).Distancedependent transfer costs
capture the biology of transfers more accurately than having
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Reconciliation using Duplication,Transfer,and Loss
a ﬁxed transfer cost (Andamand Gogarten,2011),and its use
may lead to more accurate DTL reconciliations.
In addition,we also discuss how to efﬁciently incorporate other
enhancements such as detecting transfers fromunsampled or extinct
lineages that further improve the accuracy of DTL reconciliation.
Our O(mn)time algorithm for undated species trees builds on the
O(mn)time algorithm from Toﬁgh (2009) that computes optimal
reconciliation scenarios under a simpler reconciliation cost that
ignores losses.Speciﬁcally,we showhowto augment that algorithm
to efﬁciently keep track of losses as well.Fully dated species
trees presented a greater algorithmic challenge and to obtain our
fast O(mnlogn)time algorithm,we developed a novel algorithmic
framework that exploits the structure of fully dated species trees
and makes use of recent algorithmic advances on the dynamic range
minimum query problem (Brodal et al.,2011).
Our new algorithms and other enhancements represent a great
improvement in the runtime and applicability of DTLreconciliation
compared with extant approaches.They not only make it possible
to analyze large gene families but also to quickly analyze thousands
of gene families from across the entire genomes of the species
under consideration.Furthermore,and perhaps most importantly,
they make DTL reconciliation much more amenable for use in
sophisticated applications such as reconciliationbased gene tree or
species tree reconstruction.The ability to efﬁciently handle distance
dependent transfer costs,as well as the other enhancements,in turn,
makes it possible to reconstruct the evolutionary history of gene
families even more accurately.We benchmark our algorithms to
both simulated and biological datasets and demonstrate the dramatic
improvements in runtime at a range of dataset sizes.We also assess
the accuracy of DTL reconciliation,on both dated and undated
species trees,compared with optimal tcDTLreconciliations on fully
dated trees and demonstrate the utility of using distancedependent
transfer costs in the reconciliation model.In the interest of brevity,
all proofs appear in the Supplementary Material (Section S.1).
2 DEFINITIONS AND PRELIMINARIES
Given a tree T,we denote its node,edge and leaf sets by V(T),E(T)
and Le(T),respectively.If T is rooted,the root node of T is denoted
by rt(T),the parent of a node v∈V(T) by pa
T
(v),its set of children
by Ch
T
(v),and the (maximal) subtree of T rooted at v by T(v).If
two nodes in T have the same parent,they are called siblings.The set
of internal nodes of T,denoted I(T),is deﬁned to be V(T)\Le(T).
We deﬁne ≤
T
to be the partial order on V(T),where x≤
T
y if y is a
node on the path between rt(T) and x.The partial order ≥
T
is deﬁned
analogously,i.e.,x≥
T
y if x is a node on the path between rt(T) and
x.We say that v is an ancestor of u,or that u is a descendant of v,if
u≤
T
v (note that,under this deﬁnition,every node is a descendant
as well as ancestor of itself).We say that x and y are incomparable
if neither u≤
T
v nor v≤
T
u.Given a nonempty subset L⊆Le(T),
we denote by lca
T
(L),the least common ancestor (LCA) of all the
leaves in L in tree T;that is,lca
T
(L) is the unique smallest upper
bound of L under ≤
T
.Given x,y∈V(T),x→
T
y denotes the unique
path from x to y in T.We denote by d
T
(x,y) the number of edges
on the path x→
T
y.Throughout this work,unless otherwise stated,
the term tree refers to a rooted binary tree.
Aspecies tree is a tree that depicts the evolutionary relationships
of a set of species.Given a gene family from a set of species,a
gene tree is a tree that depicts the evolutionary relationships among
the sequences encoding only that gene family in the given set of
species.Thus,the nodes in a gene tree represent genes.We assume
that each leaf of the gene trees is labeled with the species from
which that gene was sampled.This labeling deﬁnes a leafmapping
L
G,S
:Le(G) →Le(S) that maps a leaf node g∈Le(G) to that unique
leaf node s∈Le(S) which has the same label as g.Note that gene
trees may have more than one gene sampled fromthe same species.
Throughout this work,we denote the gene tree and species tree under
consideration by G and S,respectively,and will implicitly assume
that L
G,S
(g) is well deﬁned.
2.1 Reconciliation and DTLscenarios
Reconciling a gene tree with a species tree involves mapping the
gene tree into the species tree.Such a mapping allows us to infer
the evolutionary events that gave rise to that particular gene tree.
In this case,the evolutionary events of interest are speciation,
gene duplication,horizontal gene transfer and gene loss.Next,
we deﬁne what constitutes a valid reconciliation;speciﬁcally,we
deﬁne a DTL scenario (Toﬁgh et al.,2011) for G and S that
characterizes the mappings of G into S that constitute a biologically
valid reconciliation.Essentially,DTL scenarios map each gene tree
node to a unique species tree node in a consistent way that respects
the immediate temporal constraints implied by the species tree
and designate each gene tree node as representing a speciation,
duplication or transfer event.For any gene tree node,say g,that
represents a transfer event,DTLscenarios also specify which of the
two edges (g,g
) or (g,g
),where g
and g
denote the children
of g,represents the transfer edge on S,and identify the recipient
species of the corresponding transfer.
Incorporating available divergence time information.When
accurate divergence time information is available,for some or all
of the nodes of the species tree,DTL scenarios must respect the
temporal constraints imposed by the available timing information.
Speciﬁcally,those transfer events that are inconsistent with the
available timing information are disallowed (as transfer events could
only have happened between two coexisting species).If there is no
divergence time information available,then transfers are allowed to
occur between any pair of incomparable species on the species tree.
The deﬁnition of a DTL scenario below is a generalization of the
deﬁnition fromToﬁgh et al.(2011).The generalization is necessary
to correctly handle optimal reconciliations in cases when the species
tree is dated.Speciﬁcally,we enforce that,if the species tree is
dated,then transfers can only occur between coexisting species and
introduce an additional variable to explicitly specify the recipient
species for any transfer event.
Deﬁnition 2.1 (DTL scenario).A DTL scenario for G and S
is a seventuple L,M,,,,,τ
,where L:Le(G) →Le(S)
represents the leaf mapping from G to S,M:V(G) →V(S) maps
each node of Gto a node of S,the sets ,and partition I(G) into
speciation,duplication and transfer nodes,respectively,is a subset
of gene tree edges that represent transfer edges,and τ:→V(S)
speciﬁes the recipient species for each transfer event,subject to the
following constraints:
(1) If g∈Le(G),then M(g) =L(g).
(2) If g∈I(G),and g
and g
denote the children of g,then,
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M.S.Bansal et al.
(a) M(g) ≤
S
M(g
) and M(g) ≤
S
M(g
).
(b) At least one of M(g
) and M(g
) is a descendant of
M(g).
(3) Given any edge (g,g
) ∈E(G),(g,g
) ∈if and only if M(g)
and M(g
) are incomparable.
(4) If g∈I(G) and g
and g
denote the children of g,then,
(a) g∈ only if M(g) =lca(M(g
),M(g
)) and M(g
)
and M(g
) are incomparable,
(b) g∈ only if M(g) ≥
S
lca(M(g
),M(g
)),
(c) g∈ if and only if either (g,g
) ∈ or (g,g
) ∈,
(d) If g∈ and (g,g
) ∈,then M(g) and τ(g) must be
incomparable,the species represented by them must
be potentially coexisting with respect to the available
divergence time estimates,and M(g
) must be a
descendant of τ(g),i.e.M(g
) ≤
S
τ(g).
Constraint 1 above ensures that the mapping Mis consistent with
the leaf mapping L.Constraint 2(a) imposes on Mthe temporal
constraints implied by S.Constraint 2(b) implies that any internal
node in G may represent at most one transfer event.Constraint 3
determines the edges of G that are transfer edges.Constraints 4(a–c)
state the conditions under which an internal node of Gmay represent
a speciation,duplication and transfer,respectively.Constraint 4(d)
speciﬁes which species may be designated as the recipient species
for any given transfer event.
DTL scenarios correspond naturally to reconciliations and it is
straightforward to infer the reconciliation of G and S implied by
any DTL scenario.Figure 1 shows two simple DTL scenarios.
Given a DTL scenario,one can directly count the minimum
number of gene losses in the corresponding reconciliation as
follows:
Deﬁnition 2.2 (Losses).Given a DTL scenario α=
L,M,,,,,τ
for G and S,let g∈V(G) and
{g
,g
}=Ch(g).The number of losses Loss
α
(g) at node g is
deﬁned to be
• d
S
(M(g),M(g
))−1+d
S
(M(g),M(g
))−1,if g∈
• d
S
(M(g),M(g
)),if g∈ and M(g) =M(g
).
• d
S
(M(g),M(g
))+d
S
(M(g),M(g
)),if g∈,M(g) =
M(g
),and M(g) =M(g
),and
• d
S
(M(g),M(g
))+d
S
(τ(g),M(g
)) if (g,g
) ∈.
(a) (b)
Fig.1.Simple DTL scenarios.(a) and (b) depict two possible
reconciliations of G and S:the dotted arcs show the mapping M(with
the leaf mapping being speciﬁed by the leaf labels on the gene tree),and the
label at each internal node of G speciﬁes the type of event represented by
that node.The reconciliation in (a) requires two transfers and one loss and
the one in (b) requires one duplication and two losses
We deﬁne the total number of losses in the reconciliation
corresponding to the DTL scenario α to be Loss
α
=
g∈I(G)
Loss
α
(g).
Let P
,P
and P
loss
denote the costs associated with duplication,
transfer and loss events respectively.The cost of reconciling G and
S according to a DTL scenario α is deﬁned as follows.
Deﬁnition 2.3 (Reconciliation cost of a DTL scenario).Given
a DTL scenario α= L,M,,,,,τ
for G and S,the
reconciliation cost associated with α is given by R
α
=P
·+
P
·+P
loss
·Loss
α
.
Given G and S,our goal is to ﬁnd a most parsimonious
reconciliation of G and S.More formally.
Problem 1.[Most parsimonious reconciliation (MPR)] Given G
and S,the MPR problem is to ﬁnd a DTL scenario for G and S with
minimum reconciliation cost.
Based on whether the species tree is undated or fully dated,we
distinguish two versions of the MPR problem:(i) The undated MPR
(UMPR) problemwhere the species tree is undated and (ii) the fully
dated MPR (DMPR) problemwhere every node of the species tree
has an associated divergence time estimate (or there is a known total
order on the internal nodes of the species tree).We will exploit the
local structure unique to each version to develop faster algorithms
for them.
3 COMPUTING THE MOST PARSIMONIOUS
RECONCILIATION
In this section,we ﬁrst develop our fast algorithms for the U
MPR and DMPR problems and then give a simple O(mn
2
)time
algorithmfor the (general) MPR problemthat can efﬁciently handle
distancedependent transfer costs.Before we proceed,we need a
few deﬁnitions and additional notation.
Deﬁnitions:Given any g∈I(G) and s∈V(S),let c
(g,s) denote
the cost of an optimal reconciliation of G(g) with S such that g maps
to s and g∈.The terms c
(g,s) and c
(g,s) are deﬁned similarly
for g∈and g∈,respectively.Given any g∈V(G) and s∈V(S),
we deﬁne c(g,s) to be the cost of an optimal reconciliation of G(g)
with S such that g maps to s.Thus,
c(g,s) =
⎧
⎨
⎩
0 if g∈Le(G) and s=M(g),
∞ if g∈Le(G) and s =M(g),
min{c
(g,s),c
(g,s),c
(g,s)} otherwise.
Furthermore,let in(g,s) =min
x∈V(S(s))
{P
loss
·d
S
(s,x)+c(g,x)},
out(g,s) =min
x∈V(S) incomparable to s
c(g,x),and inAlt(g,s) =
min
x∈V(S(s))
c(g,x).In other words,inAlt(g,s) is the cost of an
optimal reconciliation of G(g) with S such that g may map to any
node in V(S(s));out(g,s) is the cost of an optimal reconciliation
of G(g) with S such that g may map to any node from V(S)
that is incomparable to s;and in(g,s) is the cost of an optimal
reconciliation of G(g) with S such that g may map to any node,say
x,in V(S(s)) but with an additional reconciliation cost of one loss
event for each edge on the path from s to x.
Note that the optimal reconciliation cost of G and S is simply:
min
s∈V(S)
c(rt(G),s).The equation for c(g,s) above,used in a
dynamic programming framework and coupled with methods for
computing the values of c
(g,s),c
(g,s) and c
(g,s),forms the
basis of all our algorithms.
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Reconciliation using Duplication,Transfer,and Loss
3.1 An O(mn)time algorithmfor UMPR
The following algorithmsolves the UMPR problemin O(mn) time.
Our algorithm builds on the O(mn)time dynamic programming
algorithm from Toﬁgh (2009) that computes optimal reconciliation
scenarios under a simpler reconciliation cost that ignores losses.
We compute the values c
(g,s),c
(g,s) and c
(g,s) for each
g∈V(G) and s∈V(S) by performing a nested postorder traversal
of G and S.For efﬁciency,we save and reuse as much of the
computation from previous steps as possible,and the values in(·,·),
inAlt(·,·) and out(·,·) help us in efﬁciently computing the values
c
(g,s),c
(g,s),and c
(g,s) at each dynamic programming step.
For instance,for any g∈I(G),the value of c
(g,s) is simply:
∞if s∈Le(S),and min{in(g
,s
)+in(g
,s
),in(g
,s
)+in(g
,s
)},
where {g
,g
}=Ch
G
(g) and {s
,s
}=Ch
S
(s),if s∈I(S).The values
of c
(g,s) and c
(g,s) can be similarly computed;see Steps 10
and 18 of Algorithm UReconcile for c
(g,s) and Steps 11 and
19 for c
(g,s).The nested postorder traversal ensures that when
computing the values c
(g,s),c
(g,s) and c
(g,s) at nodes g∈G
and s∈S,all the required in(·,·),inAlt(·,·),out(·,·) and c(·,·) values
have already been computed.
Algorithm UReconcile(G,S,L)
1.
for each g∈V(G) and s∈V(S) do
2.
Initialize c(g,s),c
(g,s),c
(g,s),c
(g,s),in(g,s),
inAlt(g,s),and out(g,s) to ∞.
3.
for each g∈Le(G) do
4.
Initialize c(g,L(g)) to 0,and,for each s≥
S
L(g),initialize
in(g,s) to P
loss
·d
S
(s,L(g)) and inAlt(g,s) to 0.
5.
for each g∈I(G) in postorder do
6.
for each s∈V(S) in postorder do
7.
Let {g
,g
}=Ch
G
(g).
8.
if s∈Le(S) then
9.
c
(g,s) =∞.
10.
c
(g,s) =P
+c(g
,s)+c(g
,s).
11.
If s =rt(S),then c
(g,s) =P
+min{in(g
,s)+
out(g
,s),in(g
,s)+out(g
,s)}.
12.
c(g,s) =min{c
(g,s),c
(g,s),c
(g,s)}.
13.
in(g,s) =c(g,s).
14.
inAlt(g,s) =c(g,s).
15.
else
16.
Let {s
,s
}=Ch
S
(s).
17.
c
(g,s) =min{in(g
,s
)+in(g
,s
),in(g
,s
)+
in(g
,s
)}.
18.
c
(g,s) =P
+min
⎧
⎪
⎪
⎪
⎪
⎪
⎪
⎪
⎪
⎪
⎪
⎪
⎪
⎨
⎪
⎪
⎪
⎪
⎪
⎪
⎪
⎪
⎪
⎪
⎪
⎪
⎩
c(g
,s)+in(g
,s
)+P
loss
,
c(g
,s)+in(g
,s
)+P
loss
,
c(g
,s)+in(g
,s
)+P
loss
,
c(g
,s)+in(g
,s
)+P
loss
,
c(g
,s)+c(g
,s),
in(g
,s
)+in(g
,s
)+2P
loss
,
in(g
,s
)+in(g
,s
)+2P
loss
,
in(g
,s
)+in(g
,s
)+2P
loss
,
in(g
,s
)+in(g
,s
)+2P
loss
.
19.
If s =rt(S),then c
(g,s) =P
+min{in(g
,s)+
out(g
,s),in(g
,s)+out(g
,s)}.
20.
c(g,s) =min{c
(g,s),c
(g,s),c
(g,s)}.
21.
in(g,s) =min{c(g,s),in(g,s
)+P
loss
,in(g,s
)+P
loss
}.
22.
inAlt(g,s) =min{c(g,s),inAlt(g,s
),inAlt(g,s
)}.
23.
for each s∈I(S) in preorder do
24.
Let {s
,s
}=Ch
S
(s).
25.
out(g,s
) =min{out(g,s),inAlt(g,s
)},and out(g,s
) =
min{out(g,s),inAlt(g,s
)}.
26.
Return min
s∈V(S)
c(rt(G),s).
Remarks:(i) Note that,while the above algorithm only outputs
the optimal reconciliation cost,it can be easily adapted,without
affecting its time complexity,to output the DTL scenario itself.(ii)
The algorithmabove implicitly assumes that if g∈I(G) is a transfer
node such that (g,g
) ∈,then τ(g) =M(g
).The reason for this
is easy to see:any reconciliation in which τ(g) is not M(g) (and
losses have a strictly positive cost),cannot be most parsimonious.
This,however,only holds true for the UMPR problem,and we will
be unable to make this assumption when working with partially or
fully dated species trees.
We have the following theorem.(all proofs are available in the
Supplementary Material).
Theorem 3.1.The UMPR problem on G and S can be solved in
O(mn) time.
3.2 An O(mnlogn)time algorithmfor DMPR
In the DMPR problem,there exists a total ordering of the internal
nodes of the species tree based on their divergence times.Thus,in
this setting,for any given pair of species tree edges,it is known
whether the two species represented by those edges overlapped in
their time of existence,and transfers are only allowed between two
species if they are coexisting.
We assign consecutive positive integers,starting with one,to
the internal nodes of the species tree according to the total order.
These numbers are referred to as time stamps and they represent
the temporal order in which the species represented by these nodes
diverged.Given a node s∈V(S),we denote its time stamp by t(s).
If the largest time stamp assigned to the internal nodes is k,then we
assign time stamp k +1 to each leaf of S.Any two consecutive time
stamps x,x+1 deﬁne the time zone labeled x on S.
Given a node s∈V(S)\rt(S),the species represented by that
node exists along the edge (pa(s),s) and is consequently associated
with the time stamp interval [t(pa(s)),t(s)] and the time zones
t(pa(s)),...,t(s)−1.Observe that any edge from E(S) is associated
with at least one time zone.Given any pair of nodes s,s
∈V(S)\
rt(S),a transfer is allowed between the species represented by those
nodes if and only if the two edges (pa(s),s) and (pa(s
),s
) overlap
in at least one time zone.
Our algorithm for the DMPR problem,called Algorithm D
reconcile,makes use of the same overall dynamic programming
structure as Algorithm UReconcile,and the procedure for
computing the values c
(·,·) and c
(·,·) remains identical.The
difference is in the way c
(·,·) is computed,as we can no longer
rely on the out(·,·) values.Instead,we need a more elaborate
procedure that can efﬁciently yield the ‘best receiver’ for a transfer
originating at the species tree node currently under consideration,
from among the relevant time zones.More concretely,suppose
we want to compute the value c
(g,s) assuming that (g,g
) ∈,
where g
∈Ch(g),for each s∈V(S).Our algorithm ﬁrst efﬁciently
computes the locally best and locally secondbest receivers of gene
g in each time zone based on the values c(g
,·).Then,for each
candidate node s under consideration,we efﬁciently compute the
best receiver,for a transfer originating at s,by choosing the globally
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M.S.Bansal et al.
optimal value from among the previously computed locally best
and locally secondbest receivers for the relevant time zones.For
efﬁciency,our algorithm makes use of (i) a binomial heap data
structure and (ii) a dynamic range minimum query data structure.
The binomial heap data structure maintains a set of Pvalues while
supporting ﬁndmin,insert and delete operations in O(1),O(logp)
and O(logp) time,respectively (Cormen et al.,2009;Vuillemin,
1978).The dynamic range minimumquery data structure maintains
an ordered list of numbers and can answer queries that seek the
smallest element in a given query range in O(logp) time and also
supports update operations that change the value of an element in
the list in O(logp) time (Brodal et al.,2011).
Deﬁnitions.Let k denote the number of time zones on the species
tree.Given a time zone i (1≤i ≤k),let Z(i) denote the set of edges
from E(S) that are associated with time zone i.Let Best(g,i) and
secondBest(g,i) denote,respectively,the two edges from Z(i) with
the smallest value of in(g,·).
Preprocessing.Before running Algorithm DReconcile,we assume
that we have precomputed,for each time zone i (1≤i ≤k),the
following:(i) the set of edges (pa(s),s) ∈E(S) for which t(s)−1=i
(i.e.(pa(s),s) is associated with Z(i),but not with Z(i +1)),referred
to as end(i) and (ii) the set of edges (pa(s),s) ∈E(S) for which
t(pa(s)) =i (i.e.(pa(s),s) is associated with Z(i),but not with
Z(i −1)),referred to as begin(i).
The algorithm below makes use of the procedure bestReceiver
which takes as input a node g∈I(G),a child x of g,and an edge s
from S and returns,from among all those edges that share at least
one time zone with s,an edge (pa(y),y) for which the value in(g,y)
is smallest.Essentially,the returned edge (pa(y),y) implies that,in
a scenario where g maps to s and g is a transfer node with (g,x) ∈
,the best possible mapping for x (i.e.one for which c
(g,s) is
minimized) is y.
Algorithm D−Reconcile(G,S,L)
1.
Let k denote the number of time zones on S.
2.
for each g∈V(G) and s∈V(S) do
3.
Initialize c(g,s),c
(g,s),c
(g,s),c
(g,s) and in(g,s) to ∞.
4.
for each g∈Le(G) do
5.
Initialize c(g,L(g)) to 0,and,for each s≥
S
L(g),initialize
in(g,s) to P
loss
·d
S
(s,L(g)).
6.
for each g∈I(G) in postorder do
7.
Let {g
,g
}=Ch
G
(g).
8.
for each x∈{g
,g
} do
9.
Create an empty binomial heap data structure H.
10.
Consider each edge (pa(y),y) from Z(k) and add it to H
based on the value in(x,y).
11.
Query the heap Hto assign Best(x,k) and secondBest(x,k).
12.
for each time zone i in decreasing order fromk −1 to 1 do
13.
Update the heap H by deleting from it all the edges
in begin(i +1) and inserting all the edges in end(i)
(according to their in(x,·) scores).
14.
Query the heap H to assign Best(x,i) and
secondBest(x,i).
15.
Add all the edges Best(x,·) and secondBest(x,·),labeled by
their c(x,·) scores,to a dynamic range minimumquery data
structure,indexed by their time zones (Note that,as stated,
each index gets assigned two values,which makes for an
illdeﬁned range minimumquery data structure.However,
this is easy to get around by assigning Best(x,i) to index
2i −1,and secondBest(x,i) to index 2i,and querying the
data structure accordingly).We denote this data structure
by
x
.
16.
Delete the heap H.
17.
for each s∈V(S) in postorder do
18.
If s =rt(S),then let (pa(u),u) =bestReceiver(g,g
,s),and
(pa(v),v) =bestReceiver(g,g
,s).
19.
This part of the algorithmis identical to Steps 8 through 22
of Algorithm UReconcile,except,
(a) Steps 11 and 19 are replaced by the following:
If s =rt(S),then c
(g,s) =P
+min{in(g
,s)+
c(g
,v),in(g
,s)+c(g
,u)},and,
(b) Steps 14 and 22 are removed.
20.
Delete the data structures
g
and
g
.
21.
Return min
s∈V(S)
c(rt(G),s).
Procedure bestReceiver is implemented as follows:
Procedure bestReceiver(g,x,s)
1.
Query the data structure
x
with the query range
[t(pa(s)),t(s)−1].Let e denote the returned edge.
2.
If e happens to be the edge (pa(s),s),then remove e from
x
,
and repeat the above step.
3.
Reinsert any removed edges back into
x
.
4.
Return e.
Theorem 3.2.The DMPR problem on G and S can be solved in
O(mnlogn) time.
3.3 Considering distancedependent transfer costs
Under the current reconciliation model,all transfers have the same
cost irrespective of the span of the transfer.However,it has been
observed that transfers are more likely to occur between closely
related species than between distantly related ones (Andam and
Gogarten,2011).This suggests that,ideally,the cost of a transfer
should depend on the phylogenetic distance between the donating
and receiving species.Such a cost scheme could be implemented
in several different ways:one straightforward way to implement
this is to deﬁne the transfer cost between species a and b to be
P
(a,b) =θ
1
+d
S
(a,b)·θ
2
,where θ
1
,θ
2
≥0.If branch lengths are
available on the species tree,d
S
(a,b) could also be replaced by a
term that counts the total branch length between a and b.Asimpler
alternative is to have different constant transfer costs for different
ranges of transfer spans.
Next,we give a simple O(mn
2
)time algorithm for the (general)
MPR problem that can work with undated,partially dated,or fully
dated species trees and can handle distancedependent transfer costs.
This makes it a factor of n faster than the fastest known algorithmthat
can handle distancedependent transfer costs.Our algorithm,which
we will refer to as algorithm reconcile,is essentially the same as
algorithm UReconcile,except that we remove our dependence on
the out array and assign a cost of ∞to those transfers that violate
any given time constraints.Speciﬁcally,we (i) remove Lines 14,22,
and 23 through 25 and (ii) replace Steps 11 and 19 with the
following ﬁve:
Let X ={x∈V(S):x is incomparable to and potentially
coexisting with s}.
If X =∅ then
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Reconciliation using Duplication,Transfer,and Loss
for each x∈X
Temp(x) =P
(s,x)+min{in(g
,x)+in(g
,x),
in(g
,x)+in(g
,x)}.
c
(g,s) =min
x∈X
Temp(x)
Given any a and b,the value of P
(a,b) under distancedependent
transfer costs can be computed in constant time as long as the
value d
S
(a,b) (or its equivalent in terms of branch lengths) can
be computed in constant time.This can be achieved after an O(n)
preprocessing of the species tree,which (i) allows constant time
LCAquerying (Bender et al.,2005) and (ii) labels each species tree
node with its distance (or total branch length) from the root.This
yields the following theorem.
Theorem 3.3.The MPR problem on G and S with distance
dependent transfer costs can be solved in O(mn
2
) time.
3.4 Algorithmic extensions
Unrooted gene trees.If the input gene trees are unrooted,each
possible rooted version of the unrooted gene tree is reconciled
against the species tree and the goal is to ﬁnd a reconciliation that
has minimumcost among all rootings.Each of our three algorithms
described earlier can be easily extended to work with unrooted gene
trees without any increase in their respective time complexities.
This is done by relying on the oftused observation (Chen et al.,
2000) that,w.r.t.any internal node g,all rootings of the tree can
be partitioned into three sets,depending on which of the tree edges
incident on the node is closest to the root node.We have implemented
this feature into our software RANGERDTL.
Multiple optimal solutions.It should be noted that,for any given
values of the event costs P
,P
and P
loss
,there may be more
than one optimal solution for the MPR problem.The O(mn
2
)
algorithmabove can be easily adapted to output all possible optimal
reconciliations for any given problem instance.
Further enhancements.It is also possible to extend each of our
three algorithms,without any increase in their time complexities,
to consider more complex biological scenarios,such as transfers
from potentially extinct or unsampled lineages,or transfer from
a species that then loses its copy of that gene.A more detailed
discussion of these enhancements appears in the Supplementary
Material (Section S.2).
4 EXPERIMENTAL EVALUATION
We implemented our fast algorithms into a software package called
RANGERDTL (Rapid ANalysis of Gene family Evolution using
ReconciliationDTL).Since the accuracy and utility of DTL and
tcDTL reconciliation for inferring gene family evolution have
already been demonstrated elsewhere (David andAlm,2011;Doyon
et al.,2010;Gorbunov and Liubetskii,2009;Toﬁgh,2009),we do
not attempt to do so here.Instead,our goal is to (i) demonstrate
the immense speedup in running time achieved by our algorithms
over existing stateoftheart programs;(ii) compare the solutions
obtained by DTL reconciliation on undated and fully dated species
trees against tcDTLreconciliation on fully dated trees (which can be
thought of as a ‘gold standard’);and (iii) demonstrate the utility of
enhancements such as distancedependent transfer costs.To that end,
we applied RANGERDTL to a variety of simulated and biological
datasets.Speciﬁcally,we created 500 simulated datasets (gene tree–
species tree pairs),100 each with 50,100,200,500 and 1000 taxa
generated using the probabilistic gene evolution model described
in Arvestad et al.(2009);Toﬁgh (2009);Toﬁgh et al.(2011).We
ensured that each simulated gene tree had at least one gene from
each species in the corresponding species tree,and they contained
on average 98.2,195,334.3,618.8 and 1423.5 leaves,respectively,
for the 50,100,200,500 and 1000 taxa datasets.We also created a
10 000taxon gene tree–species tree pair with random topologies
to demonstrate the feasibility of analyzing even extremely large
trees with RANGERDTL.We point out that the running time
depend only on the sizes of the input gene and species trees and
are thus independent of the actual rate parameters used to generate
the simulated trees and of the event costs used to compute the
reconciliation.Our biological dataset was derived from David and
Alm (2011) and consists of over 4700 unrooted gene trees with a
species tree of 100 (predominantly prokaryotic) species sampled
broadly across the tree of life.This biological dataset was analyzed
using the same cost parameters (P
=2,P
=3,P
loss
=1) used
in David and Alm (2011).
Running time.To compare the running time of our algorithms,we
used an implementation of our algorithm for DTL reconciliation
on undated species trees,referred to as the RANGERDTLU
program,and compared it against AnGST (David and Alm,2011)
and Mowgli (Doyon et al.,2010) which are two of the most
advanced programs implementing the fastest known algorithms
for DTL reconciliation on undated species trees and tcDTL
reconciliation on fully dated species trees,respectively.When
running RANGERDTLU and AnGST on these datasets,all
divergencetime information (branch lengths) on the nodes of the
species trees was ignored.Moreover while both RANGERDTL
and AnGST can efﬁciently handle unrooted gene trees,Mowgli
cannot;thus,we ﬁrst randomly rooted each of the 4733 gene trees
of the biological dataset.Table 1 depicts the results.We ﬁnd a
dramatic improvement in runtime and scalability over both AnGST
and Mowgli.For instance,on the 100 simulated 100taxon datasets,
RANGERDTLU is an impressive 300 and 4500 times faster than
AnGST and Mowgli,respectively.Similar speedups are observed
on the biological dataset as well,with RANGERDTLU requiring
just over a minute to analyze the entire dataset of 4733 gene trees.
(Even when run directly on the original unrooted gene trees,it
requires only about 2 min to analyze the entire dataset).Moreover,
the speedups are,as anticipated,even greater for larger datasets.
AnGST required between 8 and 10 h on each of the 10 randomly
chosen 500taxon datasets that we tried,suggesting a running time
of at least 800 h on all 100 datasets,and it crashed immediately
on the 1000taxon datasets.Similarly,Mowgli crashed after ∼4 h
of running time on each of the 10 randomly chosen 500taxon
datasets that we tried,and did not terminate in 60 h (after which
we stopped the program) on any one of the 10 1000taxon datasets
we ran it on.This suggests a total running time of at least 400 and
6000 h on all 100 of the 500 and 1000taxon datasets,respectively,
for Mowgli.In contrast,RANGERDTLU required <2 s on each
1000taxon dataset,which is,remarkably,over 100 000 times faster
than Mowgli.While neither AnGST nor Mowgli can be run on
the 10 000taxon dataset,RANGERDTLU required only ∼4 h to
analyze it.
Solution quality.Note that it is ineffective to compare the
actual reconciliations themselves as the presence of multiple
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M.S.Bansal et al.
Table 1.Runtime comparison
Dataset type Dataset size RANGERDTLU AnGST Mowgli
Simulated
50 taxa (100 datasets) 2 s 3 m:26 s 28 m:30 s
100 taxa (100 datasets) 3 s 15 m:4 s 3 h:52 m
200 taxa (100 datasets) 9 s 1 h:2 m 29 h:43 m
500 taxa (100 datasets) 35 s >800 h >400 h
1000 taxa (100 datasets) 2 m:57 s — >6000 h
10 000 taxa (1 dataset) 4 h:7 m — —
Biological 4733 gene trees,100 taxa species tree 1 m:03 s 3 h:45 m 41 h:36 m
This table shows the runtimes of RANGERDTLU,AnGSTand Mowgli on simulated and biological datasets.Times are shown in hours (h),minutes (m) and seconds (s).Experiments
were performed on a desktop computer with a 3.2 GHz Intel Core i3 processor and 4 GB of RAM.
optimal reconciliations confounds the ability to make meaningful
comparisons.Thus,we focused on comparing the reported optimal
reconciliation costs.On all datasets,the reconciliation costs reported
by RANGERDTLU are,as expected,identical to those reported
by AnGST.When compared with Mowgli,we observed that the
reconciliation costs reported by RANGERDTLUwere 7.9%lower
on the biological dataset.The fact that the costs reported by
RANGERDTLUare smaller is unsurprising as it ignores all timing
information,while Mowgli uses it.The timing information on the
biological species tree is also likely to be at least slightly inaccurate,
further contributing to the difference in reconciliation costs.On the
simulated datasets,we observed practically no difference in the
scores for RANGERDTLU and Mowgli,even on datasets with
high rates of duplication,transfer and loss (results not shown),likely
due to the fact that simulations inherently simplify the evolutionary
process and yield less complex gene trees.We also ran the fully dated
version of DTLreconciliation,RANGERDTLD,on the biological
dataset and observed that,compared with Mowgli,the reported costs
are on average only 3.7%lower.Overall,our experiments showthat
(i) on fully dated trees,solutions to the DTL reconciliation problem
closely approximate solutions obtained by tcDTLreconciliation;and
(ii) even when the species trees are undated,the DTL reconciliation
problem yields solutions that are largely similar to those obtained
with perfect timing information.
Distancedependent transfer costs.To test the utility of
incorporating distancedependent transfer costs,we modiﬁed the
RANGERDTLD program so as to increase the transfer cost by
2 over its current constant value of 3 whenever the transfer edge
spanned more than 10 edges (which represents a sizable distance in
a species tree with only 100 taxa).We observed that the reported
costs,on the biological dataset,were on average 17.2%higher than
the unmodiﬁed RANGERDTLD.This implies that the computed
optimal reconciliations contain a large number of transfer events
that span >10 edges.This strongly suggests that using distance
dependent transfer costs is likely to have a signiﬁcant impact on the
quality of the inferred reconciliations.
It is worth mentioning that even our general O(mn
2
) algorithm
for the MPR problem with distancedependent transfer costs
signiﬁcantly outperforms AnGST and Mowgli in terms of running
time.For example,on the entire biological dataset of 4733 gene
trees,it requires ∼13 min of running time,compared with almost
4 h by AnGST and over 41 h by Mowgli.Even on the 1000taxon
datasets,it required <15 min per dataset.Although we have not yet
implemented our fast O(mnlogn)time algorithm for the DMPR
problem (since the general O(mn
2
) algorithm solves the DMPR
problem as well),its runtime can be expected to be only slightly
higher than that of RANGERDTLU.
RANGERDTL can be freely downloaded from
http://compbio.mit.edu/rangerdtl/.
5 DISCUSSION AND CONCLUSION
In this article,we addressed the DTL reconciliation problem for
reconstructing gene family evolution.We proposed new algorithms
that are dramatically faster than any existing algorithms for
this problem and proposed several enhancements necessary for
improving the utility and accuracy of the computed solutions.
Our work represents a substantial improvement in the ability to
accurately analyze large gene families.It also enables,for the
ﬁrst time,the use of powerful,reconciliationbased gene tree and
species tree reconstruction methods for prokaryotes.For instance,
to reconstruct a 100taxon species tree by gene tree parsimony,using
a standard local search heuristic,one would need to reconcile on the
order of many millions of gene tree/species tree pairs;using even
the fastest existing DTL reconciliation algorithms,such as AnGST,
one would require several years of computing time to performsuch
an analysis,compared with just a few days using RANGERDTL.
There are a number of ways to further improve the accuracy of
DTLreconciliation and we would like to explore these in the future.
For instance,it would help to explicitly distinguish between two
types of transfers:ones that contribute an additional gene to the
recipient genome and those that recombine with an existing gene
copy and replace it.Under the current DTL reconciliation models,
recombining transfers are counted as a transfer followed by a loss.
Moreover,our current implementation assumes that the input gene
tree topology is correct and it would be very useful to have an
effective way to deal with any uncertainty in gene tree topologies.
ACKNOWLEDGEMENTS
The authors thank Ali Toﬁgh for help with the tree simulation
software,Lawrence David for providing the biological dataset,and
Matt Rasmussen for helpful discussions.
Funding:National Science Foundation CAREER award 0644282
to M.K.,National Institutes of Health RC2 HG005639 to M.K.and
National Science Foundation AToL 0936234 to E.J.A.and M.K.
Conﬂict of Interest:none declared.
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Reconciliation using Duplication,Transfer,and Loss
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